Rapid-Fire Probability Questions
You are in a timed quantitative screen. Answer each of the following independent probability questions, ideally in well under a minute each. For every part, state the final answer and the one-line reasoning or formula behind it — the goal is correct recall of standard structure (symmetry, complements, classic distributions), not heavy computation.
Assume fair, independent dice and coins, and a standard, well-shuffled 52-card deck (26 red, 26 black) unless a part says otherwise.
-
Two fair six-sided dice are rolled. What is the probability distribution of the sum
S
?
-
From a well-shuffled standard 52-card deck, you discard the top 10 cards
unseen
. What is the probability that the next (11th) card is red?
-
When rolling two fair six-sided dice, what is the probability that the sum is
11 or 12
?
-
In a fair-coin gambler's-ruin process you start with bankroll 10 and a goal of 20, betting 1 each round, with absorbing barriers at 0 and 20. What is the probability of reaching 20 before going broke?
-
A fair six-sided die is rolled
N
times. For a fixed face, what is the
expected number of occurrences
and the
variance
of that count?
-
Four fair six-sided dice are rolled. What is the probability that the
product
of the four outcomes is odd?
Constraints & Assumptions
-
Each die is fair and six-sided: faces
{1,2,3,4,5,6}
, each with probability
1/6
; rolls are independent.
-
The coin in part 4 is fair (
p=1/2
up,
1/2
down) and steps are
±1
.
-
The 52-card deck has exactly 26 red and 26 black cards and is uniformly shuffled.
-
"Distribution" in part 1 means the full PMF over the support, not just the mean.
-
In part 5, "a fixed face" means one specific face (e.g. how many times a 3 appears).
Clarifying Questions to Ask
-
Part 1: Do you want the full PMF (all 11 values), or a closed-form expression, or both?
-
Part 2: Are the discarded cards revealed to me, or discarded face-down/unseen? (This changes the answer.)
-
Part 4: Is the coin fair, and is each bet exactly 1 unit with hard absorbing barriers at 0 and 20?
-
Part 5: Do you want the count for a single specified face, or the joint behavior of all six face-counts?
-
General: Should I give exact fractions, decimals, or both, and how much reasoning do you want shown per answer?
What a Strong Answer Covers
-
Correct final values:
P(S=s)=366−∣s−7∣
;
21
;
121
;
21
;
E=6N, Var=365N
;
161
.
-
Recognizing the symmetry/complement shortcuts
rather than brute-forcing — especially the unseen-discard exchangeability argument and the
i/N
gambler's-ruin result.
-
Naming the underlying structure
: triangular sum distribution as a convolution; Binomial
(N,1/6)
for a face count; product-odd as a per-die independence product.
-
Quick sanity checks
: probabilities summing to 1, the symmetric tent peaking at 7, the complement
P(product even)=15/16
.
-
Speed and crisp verbal justification
— a one-liner per part, not a page of algebra.
Follow-up Questions
-
Part 4: Redo the gambler's-ruin probability for a
biased
coin with per-step win probability
p=21
. What is the general formula, and what happens as the goal moves far away?
-
Part 5: Give the
joint
distribution of all six face-counts and the covariance between two different face-counts. Why are they negatively correlated?
-
Part 1: How does the sum distribution change shape if you roll
three
dice instead of two, and why does it start to look bell-shaped?
-
Part 6: What is the probability the
sum
of four dice is odd, and why is the answer so different from the product case?